Convergence across countries and regions - digital

Convergence across countries and regions:
theory and empirics
Angel de la Fuente*
Instituto de Análisis Económico (CSIC)
January 2000
WP 447.00
Abstract
This paper surveys the recent literature on convergence across countries and regions. We discuss
the main convergence and divergence mechanisms identified in the literature and develop a simple
model that illustrates their implications for income dynamics. We then review the existing empirical
evidence and discuss its theoretical implications. Early optimism concerning the ability of a human
capital-augmented neoclassical model to explain productivity differences across economies has been
questioned on the basis of more recent contributions that make use of panel data techniques and
obtain theoretically implausible results. Some recent research in this area tries to reconcile these
findings with sensible theoretical models by exploring the role of alternative convergence mechanisms
and the possible shortcomings of panel data techniques for convergence analysis.
______________________
* This paper was prepared for the 2000 European Investment Bank Conference on Economics and Finance:
Regional development policy and convergence in the EU. I gratefully acknowledge financial support from the
European Fund for Regional Development, the Spanish Ministry of Education through CICYT grant SEC99-1189
and the European TNR "Specialization vs. diversification: the microeconomics of regional development and the
propagation of macroeconomic shocks in Europe." I would like to thank Juan Antonio Duro for his helpful
research assistance.
Correspondence to: Angel de la Fuente, Instituto de Análisis Económico, Campus de la Universidad Autónoma de
Barcelona, 08193, Bellaterra, Barcelona. Tel: 93-580-6612. Fax: 93-580-1452. E-Mail address: delafuente@cc.uab.es.
1
1.- Introducción
In the last decade or so, growth has come to occupy an increasingly important place among the
interests of macroeconomists, displacing to some extent their previous preoccupation with the
business cycle. This change is largely due to two factors. The first one is the realization that, in terms
of medium and long-term welfare, the trend is more important than the cycle -- provided the volatility
of income remains as low as it has been during the last few decades.1 The second factor is the
increasing dissatisfaction with the traditional neoclassical models that summarized the preexisting
consensus on the determinants of growth -- essentially because of their perceived inability to account
for such key features of the data as the observed increase in international inequality or the absence of
capital flows toward less developed countries.
Dissatisfaction with the received theory has motivated the search for alternatives to the traditional
neoclassical model that has driven the recent literature on endogenous growth. At the theoretical level,
numerous authors have developed a series of models in which departures from traditional
assumptions about the properties of the production technology or the determinants of technical
progress generate predictions about the evolution of the international income distribution that stand
in sharp contrast with those of neoclassical theory. Some of these models emphasize the role of
growth factors that were ignored by previous theories and generate policy implications that are
considerably more activist than those derived from the traditional models. At the empirical level,
there is also a rich literature that attempts to test the validity of the different theoretical models that
have been proposed, and to quantify the impact of various factors of interest on growth and on the
evolution of international or interregional income disparities.
This paper provides an introduction to the theoretical and empirical literature on growth and
convergence across countries and regions. It is organized as follows. Section 2 contains some general
considerations on the convergence and divergence mechanisms identified in the growth literature. In
Section 3 I develop a simple descriptive model that attempts to capture the main immediate
determinants of the growth of output and illustrates how some key properties of technology
determine the evolution of the international or interregional income distribution. Section 4 focuses on
the empirical implementation of growth models through convergence equations. Finally, sections 5
and 6 contain a brief survey of the main empirical results on convergence and a discussion of their
theoretical implications. Section 7 concludes with a brief summary and some tentative conclusions.
1 See Lucas (1987).
2
2.- Convergence and divergence in growth theory
As the reader will soon discover, the concept of convergence plays a crucial role in the literature
we will survey. Although we will eventually provide a more precise defintion of this term, we can
provisionally interpret it as short-hand for the possible existence of a tendency towards the reduction
over time of income disparities across countries or regions. Hence, we will say that there is
convergence in a given sample when the poorer economies in it tend to grow faster than their richer
neighbours, thereby reducing the income differential between them. When we observe the opposite
pattern (i.e. when the rich grow faster and increase their lead) we will say that there is divergence in
the sample.
Economic theory does not provide unambiguous predictions about the convergence or divergence
of per capita income levels across countries or regions. It does, however, identify a series of factors or
mechanisms that are capable in principle of generating either convergence or divergence. Theoretical
models based on different assumptions about the existence or relative importance of such
mechanisms can generate very different predictions about the evolution of income disparities across
territories.
At some risk of oversimplifying, we can classify growth models into two families according to
their convergence predictions.According to those in the first group, being poor is, to some extent, an
advantage. In these models the technology is such that, other things equal, poor countries grow faster
than rich ones. This does not necessarily imply the eventual elimination of inequality (other things
may not be equal), but it does mean that the distribution of relative income per capita across
territories will tend to stabilize in the long run, provided some key "structural" characteristics of the
different economies remain unchanged over time. In the second set of models, in contrast, rich
countries grow faster and inequality increases without any bound.
The source of these contrasting predictions must be sought in very basic assumptions about the
properties of the production technology at a given point in time and about the dynamics of
technological progress. A first necessary condition for convergence is the existence of decreasing
returns to scale in capital (or, more generally, in the various types of capital considered in the model).
This assumption means that output grows less than proportionally with the stock of capital. This
implies that the marginal productivity of this factor will decrease with its accumulation, reducing
both the incentive to save and the contribution to growth of a given volume of investment and
creating a tendency for growth to slow down over time. The same mechanism generates a
convergence prediction in the cross-section: poor countries (in which capital is scarcer) will grow
faster than rich ones because they have a greater incentive to save and a enjoy faster growth with the
same rate of investment. This result will be reinforced by open-economy considerations, as the flows
of mobile factors, together with international trade, will contribute to the equalization of factor prices
and domestic products per worker. Under the opposite assumption (of increasing returns in capital),
3
the preceeding neoclassical logic is inverted and we obtain a divergence prediction. In this case, the
return on investment increases with the stock of capital per worker, favouring rich countries that tend
to grow faster than poor ones, thereby increasing inequality further.
The second factor to consider in relation with the convergence or divergence of income per capita
or productivity has to do with the determinants of technological progress. If countries differ in the
intensity of their efforts to generate or adopt new technologies, their long-term growth rates will be
different. One possible objection is that the persistence of such differences is not plausible. For
instance, it may be argued that the return on technological capital should decrease with its
accumulation, just as we would expect to find for other types of capital. In this case, large differences
across countries in rates of technological investment would not be sustainable, and there would be a
tendency towards the gradual equalization of technical efficiency levels. It is far from clear, however,
that the accumulation of knowledge should be subject to the law of diminishing returns. If the cost of
additional innovations falls with scientific or production experience, for instance, the return on
technological investment may not be a decreasing function of the stock of accumulated knowledge,
and cross-country differences in levels of technological effort could persist indefinitely.
Hence, technical progress could be an important divergence factor. But there are also forces that
point in the opposite direction. As Abramovitz (1979, 1986) and other authors have pointed out, the
public good properties ot technical knowledge have an international dimension that tends to favour
less advanced countries, provided they have the capability to absorb foreign technologies and adapt
them to their own needs. The idea is simple: not having to reinvent each wheel, followers will be in a
better position to grow quickly than the technological leader, who will have to assume the costs and
lags associated with the development of new leading-edge technologies.2 The resulting process of
technological catch up could contribute significantly to convergence, particularly within the group of
industrialized countries that are in a position to exploit the advantages derived from technological
imitation.
In addition to decreasing returns and technological diffusion, the literature identifies a third
convergence mechanism that, although featured less prominently in theoretical models, is likely to be
of great practical importance. This mechanism works through structural change, or the reallocation of
productive factors across sectors. Poorer countries and regions tend to have relatively large
agricultural sectors. Given that output per worker is typically much lower in agriculture than in
manufacturing or in the service sector, the flow of resources out of agriculture and into these other
activities tends to increase average productivity. Since this process, moreover, has generally been
more intense in poor economies than in rich ones in the last few decades, it may have contributed
significantly to the observed reduction in productivity differentials across territories.
2 The idea seems to be due originally to Gerschenkron (1952) and has been developed among others by
Abramovitz (1979, 1986), Baumol (1986), Dowrick and Nguyen (1989), Nelson and Wright (1992) and Wolff
(1991).
4
In conclusion, economic theory identifies forces with contrasting implications for income
dynamics. Convergence mechanisms feature prrominently in the neoclassical and catch-up models
that dominated the literature until recently. The perceived failure of the optimistic convergence
predictions of these models, however, has motivated the search for alternatives and contributed to the
development of new theories that incorporate various divergence factors.3 Some of the pioneers of the
"endogenous growth" literature (especially Romer (1986 and 1987a and b)) focused on the possibility
of non-decreasing returns to scale in capital alone, while other authors, such as Lucas (1988), Romer
(1990) and Grossman and Helpman (1991), developed models in which the rate of technical progress
was determined endogenously and could differ permanently across countries, reflecting differences in
structural characteristics. In both cases, the theory allows for the possibility of a sustained increase in
the level of international or interregional inequality.
3.- A formal model
In this section we will develop a simple model that summarizes the key ingredients of the two
families of growth theories we have identified in the previous section. The model can generate very
different predictions about the behaviour of the international or interregional income distribution
depending on the values of certain parameters that capture assumptions about the properties of the
production technology and the determinants of the rate of technical progress. In this model, taken
from de la Fuente (1995), the evolution of the relative income of two countries or regions (a "leader"
and a "follower"), appears as the result of two processes --the accumulation of capital and
technological progress-- whose rhythm depends on the rates of investment on physical and
knowledge capital and on the speed of technological diffusion. The analysis identifies two possible
sources of divergence or rising inequality: the existence of increasing returns in reproducible factors,
and the persistence of different rates of R&D investment in the absence of technological diffusion.
Under the alternative assumptions of decreasing returns and technological catch-up, the model
predicts that the level of international or interregional inequality will tend to stabilize with the
passage of time, generating a stationary or long-term equilibrium distribution of income in which the
relative position of each territory is determined by its investment effort.
a.- Immediate determinants of the rate of growth
Economists have generally approached the study of growth with something like an aggregate
production function in mind -- that is, starting from the hypothesis that there is a stable relationship
between aggregate output on one hand and the stocks of physical inputs and technical knowledge on
the other. From this perspective, the growth of output depends on the rate of accumulation of various
productive factors and the speed of technical progress and, ultimately, on the determinants of these
3 See Romer (1986) and Lucas (1988 and 1990) among others.
5
variables, that is, on the underlying preference and technology parameters together with economic
policies and political, social and demographic factors.
For the sake of concreteness, let us consider a world in which there are only two factors of
production and one final good. Capital (K) and labour (L) are combined to produce a homogeneous
output (Y) that can be consumed directly or used as capital in the production process. We will assume
that the production technology can be adequately described by an aggregate production function of
the form
(1) Y = ΦKa(AL) 1−a = ΦALZa
where A is an index of labour-augmenting technical efficiency and K denotes a broad capital
aggregate that includes both human and physical capital. The variable Z = K/AL denotes the
capital/labour ratio in efficiency units and the coefficients a and 1-a measure the elasticity of output
with respect to factor stocks.
To allow the possibility of increasing returns in the simplest possible way, we will assume that the
term Φ, although perceived as an exogenous constant by individual agents, is in fact a function of the
form Φ = Zb that captures the external effects associated with investment.4 Under these assumptions,
output per worker, Q, is given by
(2) Q = AZ α
where α = a + b measures the degree of returns to scale in capital taking into account this factor's
indirect contribution to productivity through possible externalities.
Given equation (2), the growth of output per worker must be the result of the accumulation of
productive factors or the outcome of technical progress. Taking logarithms of (2) and differentiating
« / Q = g , 5 can be written as
with respect to time, we see that the rate of growth of output per capita Q
Q
the sum of two terms that reflect, respectively, the rate of technical progress and the accumulation of
productive factors:
(3) gQ = g a + αgz.
In the rest of this section we will explore the immediate determinants of ga and gz . Let us start
with the second factor. Denoting by s the share of investment in GDP and by δ the rate of
depreciation, the increase in the aggregate capital stock is given by the difference between investment
and depreciation, that is,
(4) K« = sY - δK = sLQ - δK
4 This specification is basically the one proposed by Romer (1986) building on Arrow (1962) to capture the
possibility that capital accumulation may generate positive spillovers. A possible justification is provided in
Romer (1987b). If there are fixed entry costs, a larger capital stock will allow an increase in the number of firms
and a finer division of labour. Increased specialization, particularly by producers of intermediate goods, could
then improve overall efficiency.
5 We use the notation x«= dx/dt for the derivative of x with respect to time, that is, the increase in its value over
an infinitesimally short period. We will denote the instantaneous growth rate of x by gx = x« / x = d ln x/dt, and
make use of the following fact: if x = y/z, then gx = gy - g z ; similarly, if x = yz, then gx = gy+ gz .
6
« dK/dt is the instantaneous increase in the capital stock. Since Z = K/AL, the growth rate
where K=
of the stock of capital per efficiency unit of labour, g z, is the difference between gk = K« / K and the
sum of the rates of technical progress (ga ) and labour force growth (n). Using (2) and (4), it is easy to
see that
(5) gz = g k - ga - n = sZ α−1 - (n+ga+δ),
where the term Z α−1 (= Q/(K/L)) is the average product of capital. Substituting this expression into
(3), we have
(6) gQ = (1-α)ga + αsZα−1 - α(n+δ).
Finally, we have to specify the determinants of the rate of technical progress, ga. We will assume
that ga is an increasing function of the fraction of GDP invested in R&D (θ) and of the opportunities
for technological catch-up, measured by the log difference (b = ln X - ln A) between a "technological
frontier" denoted by X and the country's own technological index, A:
(7) ga = γθ+ εb.
The parameters ε and γ measure, respectively, the speed of diffusion of new technologies across
countries and the productivity of R&D. We will also assume that best-practice technology improves at
a rate gx which we will take as exogenous from the perspective of each given country and assume
constant for simplicity.
Substituting (7) into (6) we finally arrive at an expression,
(8) gQ = (1-α) (γθ+ εb) + αsZα−1 - α(n+δ),
that gives the rate of growth of output per worker, gQ , as a weighted sum of two terms that capture
the immediate determinants of the rates of technical progress and capital accumulation.
b.- Dynamics
Next, we explore the implications of equation (8) for the evolution of output per worker. To study
the dynamics of the system, it will be convenient to organize the analysis in terms of the impact of two
separate processes, capital accumulation and technical progress, on the evolution of the relative
income of two countries, a "leader" and a "follower." We will show that each of these processes, by
itself, can generate either convergence or divergence in output per worker. If the technology displays
increasing returns in capital (α > 1), the rate of return on investment increases with the stock of
capital, and the system displays "explosive" behaviour. Over time, growth accelerates in each given
country, and income differences across nations increase without bound. On the other hand, when α <
1 the return on investment falls with accumulation and this implies that stocks of capital per worker
(and hence per worker income levels) tend to converge across countries, provided they all share the
same technology and other structural parameters. Similarly, the evolution of relative technical
efficiency may adopt two quite different patterns. If there is no international technological diffusion
(ε=0), the country that invests more in R&D will always have a higher rate of productivity growth. If
there is a catch-up effect, however, the technological distance between the two countries will tend to
7
stabilize at a point at which the advantage derived from the possibility of imitation is just sufficient to
offset the lower R&D investment of the follower.
To analyze in detail the dynamics induced by capital accumulation, let us recall that the growth
rate of the stock of capital per efficiency unit of labour is given by
(5) gz = sZα−1 - (n+ga+δ).
Assuming for now that the rate of technical progress, ga, is an exogenous constant, we can draw both
terms on the right-hand side of (5) as functions of Z. As shown in Figure 1, the rate of factor
accumulation, gz , is the difference between the product of the investment rate and the average
product of capital, sZα−1 , and the constant (n+g a+δ) -- and corresponds, therefore, to the vertical
distance between the two lines, as shown in the figure.6
Figure 1: Dynamics of capital accumulation
sZ α −1
gz > 0
gz > 0
n + ga + δ
gz < 0
n + ga + δ
gz < 0
sZ α −1
Z
Z*
Z*
a.- Decreasing returns in capital ( α< 1)
Z
b.- Increasing returns ( α> 1)
The two panels of Figure 1 show that the behaviour of the dynamical system described by (5)
depends crucially on the value of α. When α < 1, that is, when the neoclassical assumption of
decreasing returns holds, the return on investment decreases with the stock of capital. Hence, the term
Z α−1, is a decreasing function of Z and cuts the horizontal line given by the constant (n+ga+δ) at the
point Z* characterized by
g z = 0 ⇒ (9) Z* =
1/(1−α)
 s 
 n+ga+δ
.
From a dynamic point of view, the key finding is that under the assumption of decreasing returns the
curve sZα−1 cuts the horizontal line from above, making the growth rate of Z a decreasing function of
its level. This implies that the steady state or long-term equilibrium described by Z* is stable. Notice
that gz is positive (that is, Z increases over time) when the stock of capital per worker is small (and
therefore the return on investment is high), and negative (Z decreases over time) when Z is "large"
6 The figure ignores the fact that the rate of technical progress, g will be changing over time, causing a vertical
a
displacement of the horizontal line. It can be shown, however, that the "whole system" is stable. Asymptotically,
the horizontal line stops shifting as ga converges to the constante value g x and Z converges to Z* as shown in the
figure.
8
(larger than Z*), for in this case the volume of investment is not enough to cover depreciation and
equip newborn workers with the average stock of capital. Hence, we can interpret the steady-state
value of the stock of capital per unit of labour, Z*, as the one corresponding to a long-term
equilibrium to which the economy gradually converges for any given initial value of Z .
In summary, when α < 1 the system is stable and the stock of capital per efficiency unit of labour
converges to its stationary value, Z*. When the external effects associated with the accumulation of
capital are sufficiently strong that α >1, the situation is very different, as shown in panel b of Figure 1.
Since the return on investment, measured by Zα−1 , is now an increasing function of the stock of
capital per efficiency unit of labour, the rate of accumulation increases with Z instead of falling.
Hence, Z grows when it is larger than Z* and falls when it is smaller, moving farther and farther away
from the steady state, which must now be interpreted as a threshold for growth rather than as a longrun equilibrium.
The implications of these results for convergence are clear. Given two countries identical except in
their initial capital stocks (i.e. with access to the same technology and similar rates of investment and
population growth), the evolution of their stocks of capital and therefore of their relative incomes
depends crucially on the existence or inexistence of increasing returns to scale in capital. Under the
assumption of decreasing returns, the stock of capital per worker (and hence average productivity)
will converge to a common value. With increasing returns, on the other hand, the advantage of the
initially richer country will increase over time.
To analyze the impact of technical progress on growth and convergence it will be convenient to
work explicitly with two countries, f and l, (follower and leader). Let us define the technological
distance between leader and follower by
b lf = al - af = (al - x) - (af - x) = bf - bl
where bl and bf denote the technological distance between each of these countries and the bestpractice frontier. Observe that the evolution of the technological gap between leader and follower, blf,
satisfies the following equation:
(10) b«lf = a«l - a«f = γ(θl - θf) + ε(bl - bf) = γ(θ l - θ f) - εblf
Figure 2 displays the dynamics of this equation under two assumptions on the value of ε. When
there is no technological diffusion (ε = 0), the leading country (which by assumption invests more in
R&D) always has a higher rate of productivity growth. As a result, b« is always positive and the
lf
technological distance between leader and follower, blf , grows without bound as shown in Figure 2a.
When ε > 0, on the other hand, the line εblf is positively-sloped and cuts the horizontal line γ(θl - θf)
at a finite value of b we will denote by b *. Under this assumption, the model is stable: b« is positive
lf
lf
lf
(that is, the technological gap increases over time) when b lf is below its stationary value, blf*, and
negative (blf decreases) otherwise (see Figure 2b). Hence, the technological gap converges to a finite
value, blf*, defined by
9
b«lf = 0
⇒ (11) blf* =
γ(θl - θ f) .
ε
Figure 2: Evolution of the technological distance between leader and follower
b lf ©
εb lf
b lf ©
b lf ©< 0
γ (θ l - θ f )
γ (θ l - θ f )
b lf ©> 0
b lf ©> 0
b ∗lf
b lf
a.- No technological diffusion ( ε = 0)
b lf
b.- Catch-up with technological diffusion ( ε > 0)
In the long run, the (logarithm of the) ratio of the technical efficiency indices of the two countries
converges to a constant value that is directly proportional to the difference between their rates of
investment in R&D, and inversely proportional to the speed of technological diffusion.
Combining the results of the partial analyses undertaken so far, we can distinguish between two
cases. When the technology exhibits increasing returns in capital (α > 1) or there is no technological
diffusion (ε = 0), the model is unstable and the growth paths of the two countries diverge. If there are
decreasing returns and technological diffusion (α < 1 and ε > 0), however, the model is stable. In the
long run, the rates of growth of the two countries converge to the world rate of technical progress, gx,
and the ratio of their per capita incomes
Q l A lZ lα
=
,
Qf A Z α
f f
approaches a strictly positive constant value whose logarithm is given by:
sl(nf+gx+δ)
γ(θ l-θ f) α
,
(12) (ql - qf)* = blf* + α(z l* - zf*) =
+
ln 
ε
1-α  sf(nl+gx+δ) 
where z = ln Z and q = ln Q. Hence, there is convergence in the sense that each country approaches a
long-run equilibrium in which its income per capita, expressed as a fraction of the sample average,
remains constant over time at a level determined by its fundamentals.
This expression shows that long-run income disparities can be attributed to differences in levels of
investment in physical and technological capital and in rates of population growth. Notice, however,
that the extent to which such differences in "fundamentals" translate into long-term productivity
differentials depends on the strength of the two convergence mechanisms present in the model. For
given values of θ, s and n, the income differential across countries will be a decreasing function of the
rate of technological diffusion (ε) and the degree of returns to scale in capital (α). Hence, both
10
convergence mechanisms tend to mitigate the level of international inequality induced by crosscountry differences in fundamentals, but do not eliminate it.
4.- From theory to empirics: a framework for empirical analysis and some convergence concepts
In the previous sections we have identified two groups of theories of growth with contrasting
implications for the evolution of the international or interregional distribution of income. While
traditional neoclassical models and those that incorporate the assumption of technological catch-up
have relatively optimistic convergence implications, some endogenous growth models based on the
assumption of increasing returns and those that emphasize the endogeneity of the rate of technical
progress can generate a tendency towards the increase of income disparities across economies.
When it comes to trying to distinguish empirically between these two families of models, the
natural starting point is probably the observation that the main testable difference between them has
to do with the sign of the partial correlation between the growth rate and the initial level of income
per capita. While this correlation should be negative according to standard neoclassical models (that
is, other things equal poorer countries should grow faster), in some models of endogenous growth the
expected sign would be the opposite one. This suggests that a natural way to try to determine which
group of models provides a better explanation of the growth experience involves estimating a
convergence equation, that is, a regression model in which the dependent variable is the growth rate of
income per capita or output per worker and the explanatory variable is the initial value of the same
income indicator. The sign of the estimated coefficient of this last variable allows us in principle to
discriminate between the two sets of alternative models.
The correct formulation of the empirical model, however, requires that we control for other
variables that may affect the growth rate of the economies in the sample. As we have seen in a
previous section, neoclassical and catch-up models predict that poor countries will grow faster than
rich ones only under certain conditions. In Solow's (1956) neoclassical model, for instance, the longterm level of income is a function of the rates of investment and population growth and can, therefore,
differ across countries. In a similar vein, Abramovitz (1979, 1986) emphasizes that the process of
technological catch-up is far from automatic. Although relative backwardness carries with it the
potential for rapid growth, the degree to which this potential is realized in a given country depends
on its "social capability" to adopt advanced foreign technologies (i.e. on factors such as the level of
schooling of its population and the availability of qualified scientific and technical personnel) and on
the existence of a political and macroeconomic environment conducive to investment and structural
change.
In short, even in models where convergence forces prevail, long-term income levels can vary
across territories, reflecting underlying differences in "fundamentals." If we do not control for such
differences, the estimated relationship between growth and initial income could be very misleading.
Imagine, for instance, that the Solow model (with decreasing returns and access by all economies to a
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common technology) is the correct one, and that richer countries display on average higher rates of
investment and lower rates of population growth than poorer countries (which is why they are richer
in the first place). According to the model, these two factors would have a positive effect on the
growth rate (during the transition to the long-run equilibrium) that could conceivably dominate the
convergence effect that makes growth a decreasing function of income with other things constant. It is
clear that if we do not include the rates of investment and population growth in the equation, we
could find that the estimated coefficient of initial income is positive and conclude, erroneously, from
this fact that the predictions of the Solow model fail to hold. To put it in a slightly different way, the
problem would be that when we do not control for the determinants of the steady state, we are
actually testing the hypothesis that all economies converge to the same long-run equilibrium. The
rejection of this hypothesis, however, has no implications for the validity of the Solow model, since
this model makes no such prediction except when the economies in the sample are exactly alike.
On the basis of the preceding discussion, we can conclude that a "minimal" model for the empirical
analysis of convergence would be an equation of the form
(13) ∆yi,t = γxit - βyi,t + εit ,
where yi,t is income per capita or per worker in territory i at the beginning of period t, ∆y i,t the
growth rate of the same variable over the period, εit a random disturbance and x it a variable or set of
variables that captures the "fundamentals" of economy i, that is, all those characteristics of this
territory that have a permanent effect on its growth rate.
a.- Structural convergence equations
Many empirical studies of growth and convergence have proceeded by estimating some variant of
equation (13). In early studies the empirical specification was frequently ad hoc and only loosely tied
with the theory.7 In recent years, however, researchers have increasingly focused on the estimation of
"structural" convergence equations derived explicitly from formal models. One of the most popular
specifications in the literature is the one derived by Mankiw, Romer and Weil (MRW 1992) from an
extended neoclassical model à la Solow (1956) that would be equivalent to the one developed in
Section 3 under the assumption that the rate of technical progress is an exogenous constant common
to all countries.8 Working with a log-linear approximation to the model around its steady state, MRW
show that the growth rate of output per worker in territory i during the period that starts at t is given
approximately by the following equation:9
7 See for instance Kormendi and McGuire (1985), Grier and Tullock (1989) and Barro (1991).
8 That is, the specification of the rate of technical progress as a function of R&D expenditure and the
technological gap is abandoned, being replaced by the simple assumption that the rate of technological progress
is an exogenous constant, g, equal for all countries. The part of the model that describes capital accumulation, on
the other hand, would be exactly as developed in Section 3.
9 Barro and Sala i Martin (1990, 1992) derive a similar expression from a variant of the optimal growth model of
Cass (1965) and Koopmans (1965) with exogenous technical progress. The resulting equation is similar to (14)
except that the investment rate (which is now endogenous) is replaced by the rate of time discount among the
determinants of the steady state. The convergence coefficient, β, is now a more complicated function of the
parameters of the model, but it still depends on the degree of decreasing returns to capital and on the rates of
population growth, depreciation and technical progress. A second difference between the two models is that,
12
sit
α
(14) ∆yi,t = g + β(aio + gt) + β
ln
- βyi,t
1-α δ+g+nit
where
(15) β = (1-α) (δ+g+n),
g is the rate of technical progress, δ the depreciation rate, α the coefficient of capital in the aggregate
production function, t the time elapsed since the beginning of the sample period, aio the logarithm of
the index of technical efficiency at time zero, s the share of investment in GDP and n the rate of
growth of the labour force.
It is important to understand that the estimation of equation (14) does not imply that we are
literally accepting the assumptions of the underlying Solow-type model (i.e. we do not need to
assume that the investment rate is exogenous or constant over time). What we are doing is simply
assigning to some of the parameters of the Solow model (in particular, to s and n) the observed
average values of their empirical counterparts during a given period. During this period, the economy
will behave approximately as if it were approaching the steady state of the Solow model that
corresponds to the contemporaneous parameter values. In the next period, of course, we are likely to
observe different values of the investment and population growth rates and therefore, a different
steady state, but this poses no real difficulty. In essence, all we are doing is constructing a convenient
approximation to the production function that allows us to recover its parameters using data on
investment flows rather than factor stocks. This is very convenient because such data are easier to
come by and can be expected to be both more reliable and more comparable over time and across
countries than most existing estimates of factor stocks. It must be kept in mind, however, that the only
information we can extract from the estimation of a convergence equation of the form (14) concerns
the properties of the production technology. As Cohen (1992) emphasizes, the estimated equation
does not, in particular, tell us anything about the actual dynamics of the economy or the position of a
hypothetical long-run equilibrium -- although it does allow us to make predictions about long-term
income levels conditional on assumptions about the future behaviour of investment and population
growth rates.
The empirical implementation of equation (13) or (14) does not, in principle, raise special problems.
Given time series data on income, population and investment for a sample of countries or regions, we
can use (14) to recover estimates of the rate of convergence and the parameters of the production
function. The convergence equation can be estimated using either cross-section or pooled data. Most
of the earlier convergence studies took the first route, averaging the variables over the entire sample
period and working with a single observation for each country or region. The second possibility,
which has become increasingly popular, involves averaging over shorter subperiods in order to obtain
several observations per country.
whereas the MRW model can be easily extended to incorporate investment in human capital, Barro and Sala do
not include this factor as an argument of the production function, although they do bring it into their empirical
specification, in an ad hoc way, as a determinant of the steady state.
13
In either case, one difficulty which immediately becomes apparent is that three of the variables on
the right-hand side of the equation (g, δ and aio ) are not directly observable. In the first two cases, the
problem is probably not very important. Although these coefficients can be estimated inside the
equation (and this has been done occasionally), the usual procedure in the literature is to impose
"reasonable" values of these parameters prior to estimation. The standard assumption is that g = 0.02
and δ = 0.03, but researchers report that estimation results are not very sensitive to changes in these
values.
The possibility that initial levels of technical efficiency (aio) may differ across countries does raise a
more difficult problem. Although some authors have argued that it may be reasonable to assume a
common value of aio because most technical knowlege is in principle accessible from everywhere,
casual observation suggests that levels of technological development differ widely across countries. If
this is so, failure to control for such differences (or for any other omitted variables) will bias the
estimates of the remaining parameters whenever the other regressors in the equation are correlated
with the missing ones. In other words, we can only legitimately subsume technological differences
across countries in the error term if they are uncorrelated with investment rates and population
growth. This seems unlikely, however, as the level of total factor productivity is one of the key
determinants of the rate of return on investment.
The standard solution for this problem is to turn to panel data techniques in order to control for
unobserved national or regional fixed effects. The simplest procedure involves introducing country or
regional dummies in order to estimate a different regression constant for each territory. It should be
noted, however, that this is equivalent to estimating the equation with the dependent and
independent variables measured in deviations from their average values (computed over time for
each country or region in the sample). Hence, this procedure (as practically all panel techniques
designed for removing fixed effects), ignores the information contained in observed cross-country
differences and produces parameter estimates which are based only on the time variation of the data
within each territory over relatively short periods. Since what we are trying to do is characterizing the
long-term dynamics of a sample of economies, this may be rather dangerous, particularly when the
data contain an important cyclical component or other short-term noise.
The structural convergence equation methodology has some important advantages and
limitations, both of which are derived from the close linkage between theory and empirics that
charaterizes this approach. Its most attractive feature is that it allows us to use the relevant theory to
explicitly guide the formulation of the empirical model -- that is, the formal model is used to
determine what variables must be included in the regression and how they must enter in order to
obtain direct estimates of the structural parameters of the model. It is clear, however, that such
guidance comes at a price, as our estimates will be, at best, only as good as the underlying theoretical
model. Hence, an inadequate specification of this model can yield very misleading conclusions.
14
Although this problem arises to some extent whenever we run a regression, there are reasons to
think that it may be particularly important in the present context. In most of the recent empirical work
on growth and convergence, the theoretical model of reference is some version of the one-sector
neoclassical model with exogenous technical progress that underlies equation (14). Since the only
convergence force present in this model is what we may call the neoclassical mechanism, the usual
finding of a negative partial correlation between growth and initial income must be interpreted in this
framework as evidence that the aggregate production function displays decreasing returns to scale in
reproducible factors. In fact, this assumption is precisely what allows us to draw inferences about the
degree of returns to scale from the estimated value of the convergence coefficient. The problem, of
course, is that if there are any other operative convergence mechanisms, the inference will not be
valid, as the estimated value of the convergence parameter will also capture their effects.
As we have seen, the literature identifies at least two factors other than decreasing returns that can
generate a negative partial correlation between income levels and growth rates holding investment
and population growth constant: technological diffusion and structural change. Although none of
these convergence mechanisms is incompatible with the neoclassical story, the observation that this is
not the only possible source of convergence suggests that it may be dangerous to accept without
question an interpretation of the convergence coefficient based too literally on the preceding model.
For instance, if income per capita is highly correlated with the level of technological development, the
coefficient of initial income in a convergence regression could capture, at least in part, a technological
catch-up effect. To avoid the danger of drawing the wrong conclusions about the properties of the
technology, it may be preferable to interpret existing estimates of the convergence parameter, β,
(particularly in the case of unconditional convergence equations) as summary measures of the joint
effect of several possible convergence mechanisms. The value of this parameter (i.e. the partial
correlation betwen the growth rate and initial income) will depend on the coefficient of capital in the
production function, the speed of technological diffusion, the impact of sectoral change and on the
response of investment rates to rising income), and will be positive (i.e. growth will be negatively
correlated with initial income) whenever the forces making for convergence dominate those working
in the opposite direction.
b. Some convergence concepts
Before we proceed to review the empirical evidence, it is convenient to introduce some concepts of
convergence that will feature prominently in the discussion below. Perhaps the first question that
arises concerning the evolution of the distribution of income per capita is whether the dispersion of
this variable (measured for instance by the standard deviation of its logarithm) tends to decrease over
time. The concept of convergence implicit in this question, called σ-convergence by Barro and Sala i
Martin (1990, 1992), is probably the one closest to the intuitive notion of convergence. It is not,
however, the only possible one. We may also ask, for instance, whether poorer countries tend to catch
15
up with richer ones, or whether the relative position of each country within the income distribution
tends to stabilize over time. The concepts of absolute and conditional β-convergence proposed by Barro
and Sala i Martin (B&S) correspond roughly to these two questions.
To make more precise these two notions of convergence, we can use a variant of equation (13) in
which we assume that each economy's fundamentals remain constant over time (that is, that x it = xi
for all t) and we interpret the variable y it as relative income per capita, that is, income per capita
normalized by the contemporaneous sample average. Omitting the disturbance term, the evolution of
relative income in territory i is described by
(13') ∆yi,t = γxi - βyi,t.
Setting ∆yi,t equal to zero in this expression, we can solve for the steady-state value of relative income,
γx
(16) yi * = i .
β
It is easy to check that if β lies between zero and one, the system described by equation (13') is stable.
This implies that the relative income of territory i converges in the long run to the equilibrium value
given by y i *. Notice that the equilibrium can differ across countries as a function of the
"fundamentals" described by xi.
In terms of this simple model, we will say that there is conditional β-convergence when β lies
between zero and one, and absolute β-convergence when this is true and, in addition, xi is the same
for all economies -- i.e. when all countries or regions in the sample converge to the same income per
capita.
Even though they are closely related, the three concepts of convergence are far from being
equivalent. Some type of β-convergence is a necessary condition for sustained σ-convergence, for the
level of inequality will grow without bound when β is negative (i.e. when the rich grow faster than
the poor). It is not sufficient, however, because a positive value of β is compatible with a transitory
increase of income dispersion due either to random shocks or to the fact that the initial level of
inequality is below its steady-state value (as determined by the dispersion of fundamentals and the
variance of the disturbance). The two types of β-convergence, moreover, have very different
implications. Absolute β convergence implies a tendency towards the equalization of per capita
incomes within the sample. Initially poor economies tend to grow faster until they catch up with the
richer ones. In the long run, expected per capita income is the same for all members of the group,
independently of its initial value. As we know, this does not mean that inequality will disappear
completely, for there will be random shocks with uneven effects on the different territories. Such
disturbances, however, will have only transitory effects, implying that, in the long run, we should
observe a fluid distribution in which the relative positions of the different regions change rapidly.
With conditional β-convergence, on the other hand, each territory converges only to its own steady
state but these can be very different from each other. Hence, a high degree of inequality could persist,
even in the long run, and we would also observe high persistence in the relative positions of the
16
different economies. In other words, rich economies will generally remain rich while the poor
continue to lag behind.
It is important to observe that, although the difference between absolute and conditional
convergence is very sharp in principle, things are often much less clear in practice. In empirical
studies we generally find that a number of variables other than initial income enter significantly in
convergence equations. This finding suggests that steady states differ across countries or regions and,
therefore, that convergence is only conditional. It is typically the case, however, that these
conditioning variables change over time and often tend to converge themselves across countries or
regions. Hence, income may still converge unconditionally in the long run, and this convergence may
reflect in part the gradual equalization of the underlying fundamentals. In this situation, a conditional
and an unconditional convergence equation will yield different estimates of the convergence rate.
There is, however, no contradiction between these estimates once we recognize that they are
measuring different things: while the unconditional parameter measures the overall intensity of a
process of income convergence which may work in part through changes over time in various
structural characteristics, the conditional parameter captures the speed at which the economy would
be approaching a "pseudo steady state" whose location is determined by the current values of the
conditioning variables.
5.- Convergence across countries and regions: empirical evidence and theoretical implications
Having reviewed the theoretical and empirical framework used in the convergence literature, we
are now in a position to examine the available empirical evidence an discuss its implications. We will
begin this section with a review of some of the more significant empirical results in this literature.
Although we will pay special attention to the case of Spain, the evolution of the regional income
distribution follows a similar pattern in most of the existing samples. In most industrial countries we
observe a significant reduction of the level of regional inequality over the medium and long run,
although this process of convergence seems to cease or at least slow down in recent years. There is
also clear evidence of β convergence: the correlation between initial income and subsequent growth is
generally negative in regional samples even without conditioning on additional variables. At the
national level, the situation is quite different. In broad country samples, the level of inequality
increases over time and beta convergence emerges only when we condition on variables like human
capital indicators and investment rates. On the other hand, the convergence rate estimated after
controlling for these variables is quite similar to the one obtained with regional samples.
In addition to their descriptive interest, these results have interesting theoretical implications. The
consensus view in the literature (at least until recently) seems to be that the apparent slowness of the
process of convergence can be taken as an indication that the production technology displays almost
constant returns to scale in capital -- a conclusion that only seems plausible if we extend the
traditional concept of capital to incorporate educational investment. Hence, the empirical results seem
17
to point towards an extended version of the neoclassical model built around a richer concept of capital
than the one we find in the traditional theory. Some recent studies, however, suggest that it is
probably premature to conclude that such a simple model provides a satisfactory description of the
growth process and of the determinants of income levels.
a.- Some "classical" results on convergence
In this section we will review some representative results of a series of studies that follow what
Sala i Martin (1995) calls the "classical approach" to convergence analysis. To summarize the key
features of the convergence pattern within a given sample, we will make use of two techniques that
have been frequently used in the literature. The first one, designed for the study of sigma
convergence, involves plotting the time path of some measure of dispersion of income per capita,
typically the standard deviation of its logarithm. To analyze the pattern of beta convergence, we
estimate an unconditional convergence equation -- i.e. a version of equation (13) without conditioning
variables in which we impose the assumption of a common intercept-- and plot the estimated
regression line together with the corresponding scatter plot, identifying each of the observations. This
procedure allows us to visualize the initial position of each economy and its performance relative to a
hypothetical average region whose behaviour is described by the fitted regression line.
Figure 3: σ convergence in the Spanish regions
0,35
0,33
0,31
0,29
0,27
0,25
0,23
0,21
0,19
0,17
0,15
1955
1962
1969
1975
1981
1987
- Note: the original income variable is regional gross value added per capita in 1990 ptas., taken from Banco
Bilbao Vizcaya (various years).
The case of Spain provides a representative illustration of what we find in most available regional
samples. Figure 3 shows the time path of the standard deviation of relative regional income per capita
(defined as log income per capita measured in deviations from its interregional average) during the
period 1955-91. The pattern of sigma convergence is clear: over the period as a whole, the standard
18
deviation of relative income per capita falls by approximately 40%. The level of inequality, however,
stabilizes after the second half of the 70s. Although this may be an indication that the regional income
distribution is close to its steady state, it may still be too soon to rule out the possibility that the
interruption of the convergence process may be a transitory phenomenon due to the oil shocks and
other macroeconomic turbulences of the last decades.
Figure 4: β convergence in the Spanish regions, 1955-91
tasa de crecimiento de la renta relativa
10
C-M
Mur
Ext
Bal
Gal
Ara
And
0
Cana
Val
CyL
Nav
Mad
Rio
Cat
Ast
Cant
-10
PV
-20
-600
-400
-200
0
200
400
600
800
renta relativa inicial
- Note: The estimated equation is
∆yi = -0,01506*yi,o t = 5,72
R 2 = 0,6859.
Figure 4 summarizes the results of an unconditional convergence regression in which the
dependent variable is the average growth rate of relative income during the whole sample period. The
negative slope of the fitted regression line indicates that, on average, growth has been faster in the
initially poorer regions. The fit of the regression is fairly good but the rate of convergence (i.e. the
slope of the regression line) suggests that the process of convergence is very slow. The value of this
coefficient (0.015) indicates that, in the case of a "typical region," only 1.5% of the income differential
with respect to the regional average is eliminated each year.
Moving on to other countries, the pattern of σ convergence at the regional level is very similar in
most industrial economies. The States of the US, the Japanese prefectures and the regions of the
European Union all display a gradual reduction of the level of inequality, although this process is
sometimes interrupted by shocks such as WWII, the Great Depression or the oil shocks. In the last two
decades, moreover, the pace of convergence slows down. The level of inequality stabilizes and even
displays a slight increase in some cases. As an illustration, Figure 5 shows the evolution of the
dispersion of personal income per capita in the states of the US during the last century.
19
Figure 5: σ convergence across the US states
1,2
1
0,8
0,6
0,4
0,2
0
1880
1900
1920
1930
1940
1950
1960
1970
1980
1988
- Note: coefficient of variation of the logarithm of personal income per capita. Source: Barro and Sala (1991).
We also find a similar pattern of beta convergence in most regional samples. Table 1 summarizes
the results of the estimation of a standard convergence equation with regional data for a number of
different countries.10 In the European case, the data for the different countries are pooled and a
common value of β is imposed with income measured in deviations from national means. Hence, the
results refer to the speed of regional convergence within each country, just as in the individual
regressions for the five largest EU members also reported in the table.
Two alternative estimates of β are reported for most samples. The first one comes from a crosssection regression of the average growth rate of income per capita over the entire sample period on
the initial level of income. The second equation is estimated with pooled data for shorter subperiods,
imposing a constant value of β but including fixed time effects. Most of the equations include as
regressors indices of the sectoral composition of output (typically the share of agriculture) in order to
control for aggregate shocks that may be correlated with initial income. In all cases, the estimated
value of the convergence parameter is positive, indicating that poorer regions tend to grow faster than
richer ones. A second empirical regularity (to which we will return in the next section) is that the
estimated value of β is very small (around 2% per year) and rather stable across samples.
It is interesting to note that results obtained with national data are slightly different. When no
additional variables are included to control for possible differences across national steady states,
10 This table is taken from a recent paper by Sala i Martin (1996) that summarizes the results of various studies
on regional convergence (in particular, Barro and Sala i Martin (1990) and (1991) for the US and several European
countries, Coulombe and Lee (1993) for Canada and Shioji (1996) for Japan. Similar results are also reported by
Dolado et al (1994) for Spain and by Svensson (1997) for Sweden.
20
divergence (i.e. a negative value of β) is the norm in large samples. When we control for educational
levels and other variables that may be considered reasonable proxies for the steady state, the
hypothesis of (conditional) convergence is accepted in all samples and the estimated convergence rate
approaches again the ubiquitous 2% figure.
Table 1: Regional convergence in different samples
___________________________________________________________________
a single long period
panel
2
β
R
β
sample and period:
(s.e.)
(s.e.)
48 US states
1880-1990
(0.002)
0.017
47 japanese prefectures
1955-1990
(0.004)
90 EU regions
1950-1990
0.015
0.018
(0.002)
(0.003)
11 German regions
1950-1990
(0.005)
11 UK regions
1950-1990
(0.007)
21 French regions
1950-1990
(0.004)
20 Italian regions
1950-1990
(0.003)
17 Spanish regions
1955-1987
(0.007)
10 Canadian provinces
1961-1991
(0.008)
0.019
0.014
0.03
0.016
0.010
0.023
0.024
0.89
0.022
(0.002)
0.59
0.031
(0.004)
0.55
0.016
(0.006)
0.61
0.029
(0.009)
0.55
0.015
(0.003)
0.46
0.016
(0.003)
0.63
0.019
(0.005)
0.29
___________________________________________________________________
- Note: standard errors in parentheses below each coefficient.
Table 2, taken from B&S (1992a) summarizes the results of the estimation of a convergence
equation with cross-section data for three different samples over roughly the same period: a broad
sample of 98 countries, a smaller one formed by the 20 original OECD members, and a third one
which comprises the 48 continental states of the US. As can be seen in the table, the results are very
different in the three cases. When we do not control for other variables, the estimated value of the
convergence parameter (β) is negative in the largest sample (equation [1]), indicating a tendency for
rich countries to grow faster than poor ones. The coefficient is positive in the other two samples
(equations [3] and [5]), but the estimated speed of convergence is twice as large in the sample of US
states than in the OECD sample.
21
Table 2: Convergence among countries and regions
________________________________________________________
Sample and period
BETA
Other
R2
[s.e.]
variables
[1] 98 countries
1960-85
-0.0037
[0.0018]
0.04
no
[2] 98 countries
1960-85
0.0184
[0.0045]
0.52
yes
[3] OECD
1960-85
0.0095
[0.0028]
0.45
no
[4] OECD
1960-85
0.0203
[0.0068]
0.69
yes
[5] 48 US states
1963-1986
0.0218
[0.0053]
0.38
no
[6] 48 US states
0.0236
0.61
yes
1963-1986
[0.0013]
________________________________________________________
- Source: Barro and Sala i Martin (1992a).
- The "other variables" included in regressions [2] and [4] are the primary and secondary enrollment rates in 1960,
public consumption (excluding defense and education) as a fraction of GDP, the average annual number of
political murders, the average number of revolutions and coups, and an index of the relative price of capital
goods (constructed by Summers and Heston) in 1967.
In addition to the initial level of income, equation [6] includes as regressors a set of regional dummies, a sectoral
composition variable and the fraction of the labour force with some university enducation in 1960.
Barro and Sala i Martin interpret these results as an indication of the relative importance of the
within-sample differences in steady states. As the sample becomes more and more homogeneous, the
bias induced in the estimation of β by the omission of the relevant control variables will decrease. The
results of equations [2], [4] and [6], where additional control variables are included, are consistent
with this interpretation. Regressions [2] and [4] include as explanatory variables a proxy for the initial
level of human capital, two indices of political stability, the share of non-productive public
expenditure in GDP, and a measure of the distortions that affect the relative price of capital goods.
Controlling for these variables, the estimated value of β is positive in both samples and very close to
the value of 2% estimated in equation [5] for the continental US states. On the other hand, the
inclusion of additional control variables (regional dummies, an index of education and a sectoral
composition variable) in the last equation increases only slightly the estimated rate of convergence
among the US states.
b.- Theoretical implications: a revised neoclassical consensus?
The papers we have just reviewed highlight three interesting empirical regularities. i) First,
evidence of some sort of beta convergence is found in practically all available samples. While
convergence is only conditional at the national level, in most regional samples a negative correlation
between initial income and subsequent growth emerges without controlling for other variables. This
22
second result is consistent with the existence of absolute convergence at the regional level -- but most
of the studies we have reviewed do not explicitly test this hypothesis. 11 ii) Second, we have seen that
the process of convergence seems to be extremely slow. Many of the existing estimates of the
convergence parameter cluster around a value of 2% per year which implies that it takes around 35
years for a typical region to reduce its income gap with the national average by one half. Hence, the
expected duration of the convergence process must be measured in decades. iii) Finally, it is
interesting to oberve that the estimated convergence coefficient is remarkably stable across samples.
This stability suggests that the mechanisms that drive convergence in income per capita across
different economies seem to operate in a regular fashion. Hence, we can at least hope to provide a
unified structural explanation of the convergence process in terms of a "general" theoretical model.
Perhaps the dominant view in the literature is that a good candidate for this "general" model is a
simple extension of the one-sector neoclassical model with exogenous technical progress. Just about
the only departure from the traditional assumptions required in order to explain the empirical
evidence is a broadening of the relevant concept of capital in order to include investment in
intangibles such as human and technological capital. This conclusion is reached essentially by
interpreting the results we have just reviewed within the framework of the growth model underlying
the conditional convergence equation given in (14). According to our previous discussion, the finding
of (at least conditional) beta convergence in most national or regional samples can be interpreted as
evidence in favour of the neoclassical assumption of decreasing returns to capital, as this result would
not be consistent with increasing returns models that predict an explosive behaviour of income and its
distribution. On the other hand, the apparent slowness of the convergence process does suggest that
we are not that far from having constant returns in reproducible factors -- a result that seems
considerably more plausible if we think in terms of a broad capital aggregate, rather than the rather
restrictive concept of capital we find in old-fashioned neoclassical models.
Since this broader concept of capital is probably one of the most significant contributions of the
recent literature to our understanding of the mechanics of growth, the issue probably deserves a fairly
detailed discussion. The reader will recall that within the framework of the Solow model the
convergence coefficient (β) depends on the degree of returns to scale, measured by α (with α = a + b,
where a is the coefficient of capital in the "private" production function and b captures the possible
externalities), and on the rates of technical progress (g), population growth (n) and depreciation (δ).
More specifically, we have seen that the relationship among these variables is given by
(15') β = (1-a-b)(δ+g+n).
11 Those that do test it by including different sets of conditioning variables generally reject it, as the significance
of many of these variables implies important cross-regional differences in steady states. (See for instance Dolado
et al (1994) and Mas et al (1995) for the Spanish provinces, Herz and Röger (1996) for the German
Raumordnungsregionen, Grahl and Simms (1993), Neven and Gouyette (1995) and Faberberg and Verspagen
(1996) for various samples of European regions, Holtz-Eakin (1993) for the states of the US and Paci and Pigliaru
(1995), Fabiani and Pellegrini (1996) and Cellini and Scorcu (1996) for the regions of Italy.) As we have noted in
Section 4b, however, this evidence does not conclusively reject the hypothesis of absolute convergence, as
conditioning variables (and hence steady states) may themselves be converging over time.
23
Using this expression and making reasonable guesses about the values of some of the parameters, we
can extract information about key properties of the production technology from empirical estimates of
the convergence rate. For a start, let us consider the expected value of β under conventional
assumptions about the values of the remaining parameters. Within the framework of a traditional
neoclassical model (with constant returns to scale in capital and labour, perfect competition and no
externalities) we would have b = 0 and a would be equal to capital's share of national income, which is
around one third. The average rate of population growth in the industrial countries during the postWWII period is approximately 1%. Available estimates of the rate of technical progress are around 2%
per year. Finally, estimates of the rate of depreciation vary considerably. In the convergence literature
it is commonly assumed that δ = 0.03, but a higher value (around 5 or 6% per year) may be more
reasonable. Given these assumptions, the expected value of β lies between 0.04 and 0.06.
As we have seen, the empirical results of Barro and Sala i Martin (1990, 1992), Mankiw, Romer and
Weil (1992) and other authors point towards a much lower convergence rate. Since the estimated
value of the parameter is still positive, the evidence is consistent with decreasing returns to capital (i.e.
a + b < 1). The low value of β, however, suggests that we are relatively close to having constant
returns to capital. Maintaining our previous assumptions about the values of the remaining
parameters, a convergence coefficient of 0.02 would imply a value of a+b between 0.67 and 0.78 -more than twice the share of capital in national income.
One possible explanation (Romer, 1987b) is that this result may reflect the existence of important
externalities associated with the accumulation of physical capital. While these external effects would
not be sufficiently strong to generate increasing returns in capital alone, they might still account for
the apparent slowness of convergence. Other authors, however, argue that a more plausible
explanation is that the omission of variables which are positively correlated with investment in
physical capital may bias upward the coefficient of this variable. Barro and Sala i Martin (1990, 1992)
argue that a value of capital's coefficient around 0.7 only makes sense if we count accumulated
educational investment as part of the stock of capital.
Mankiw, Romer and Weil (1992) advance the same hypothesis and test it explicitly by estimating a
structural convergence equation similar to equation (14) above that explicitly incorporates a proxy for
the rate of investment in human capital as a regressor. Their results, and those obtained by other
authors who estimate similar specifications,12 tend to confirm the hypothesis that investment in
human (and technological) capital plays an important role in the growth process.13 As Mankiw (1995)
points out, once human capital is included as an input in the production function, the resulting model
is consistent with some of the key features of the data. Countries that invest more in physical capital
and education tend to grow faster and therefore eventually attain high levels of relative income.
12 See for instance Lichtenberg (1992), Holtz-Eakin (1993), Nonneman and Vanhoudt (1996) and de la Fuente,
(1998b).
13 See de la Fuente (1997) for a more detailed review of this literature.
24
Cross-country differences in rates of accumulation, moreover, are sufficiently high to explain the bulk
of the observed dispersion of income levels and growth rates.
6.- Loose ends and recent developments
We have seen in the previous section that the main theoretical conclusion drawn from the earlier
studies of convergence is that a modified version of the aggregate neoclassical model provides a
satisfactory description of the process of growth and of the evolution of the regional (or national)
income distribution. The main change relative to the more traditional models is the broadening of the
relevant concept of capital in order to include human and possibly technological capital. Other than
this, the model is essentially Solow's (1956) model with exogenous technological progress and does
not incorporate any convergence mechanisms other than the one derived from the existence of
decreasing returns to capital.
It is probably fair to say that just a few years ago this extended neoclassical model summarized a
consensus view on the mechanics of growth that was shared (possibly with some reservations) by
most researchers working in this field. In recent years, however, this emerging consensus has been
challanged by a series of papers that, relying on panel data techniques, obtain results that are difficult
to reconcile with the prevailing theoretical framework. In this section we will summarize some of the
key findings of these studies and discuss the theoretical difficulties they raise.
a.- Convergence and panel data
One of the key findings of the "classical" convergence studies is that convergence to the steady
state is an extremely slow process. It has recently been argued, however, that this result may be due to
a bias arising from the use of econometric specifications that do not adequately allow for unobserved
differences across countries or regions. To get around this problem, a number of authors have
proposed the use of panel techniques that allow for unobserved fixed effects. As we will see in this
section, their results raise some puzzling questions.
Marcet (1994), Raymond and García (1994), Canova and Marcet (1995), de la Fuente (1996), Tondl
(1997) and Gorostiaga (1998), among others, estimate fixed-effects convergence models using panel
data for a variety of regional samples. Their results suggest a view of the regional convergence
process that stands in sharp contrast with the one advanced in earlier studies by B&S and other
authors: instead of slow convergence to a common income level, regional economies within a given
country seem to be converging extremely fast (at rates of up to 20% per year) but to very different
steady states.14 Cross-national studies provide a roughly similar picture: Knight et al (1993), Canova
and Marcet (1995), Islam (1995) and Caselli et al (1996), among others, find evidence of rapid
convergence across countries (at rates of up to 12% per annum) toward very different steady states
whose dispersion can be explained only in part by observed cross-national differences in population
14 Similar results are also reported by Evans and Karras (1996) for a sample of US states using time series
techniques.
25
growth and investment rates. In both cases, many of the standard conditioning variables (such as
human capital indicators) lose their statistical significance, the estimated coefficient of physical capital
adopts rather low values, and the size and significance of the regional or national fixed effects
suggests that persistent differences in levels of technical efficiency play a crucial role in explaining the
dispersion of income levels.
I will illustrate the sharp contrast between fixed-effects and pooled data or cross-section estimates
of the convergence coefficient using data for two samples of (European and Spanish) regions. For each
sample I estimate two versions of the following convergence equation
(17) ∆yrt = αr - βyrt + εrt
where ∆yrt is the average annual growth rate of relative income over the subperiod starting at time t
and α r a region-specific constant that can be used to recover an estimate of the steady-state income
level (y r * = α r /β). First, we estimate a restricted or unconditional version of equation (17) with the
pooled data after imposing the assumption of a common intercept (and therefore a common steady
state) for all regions. Next, we estimate an unrestricted or conditional version of the same equation
using ordinary least squares with dummy variables (LSDV) to estimate regional fixed effects. Finally,
we repeat the exercise using Arellano's (1988) orthogonal deviations (OD) procedure in order to try to
avoid the short sample bias that may affect LSDV estimates.
Table 3: Estimated regional convergence rates and long-term dispersion of income per capita
with various specifications
______________________________________________________________________________
[1]
[2]
[3]
[4]
[5]
[6]
β
0.022
0.080
0.076
0.0085
0.2591
0.3912
(4.76)
(5.63)
(3.91)
(14.64)
(8.83)
(t)
(3.24)
s.e. regression
0.0207
0.0201
0.0240
0.0219
std dev iation yr *
[0.000]
0.2057
0.2056
[0.000]
0.2322
0.2328
_
0.0995
0.2120
σy
σy (1993)
0.1980
0.1980
0.1980
0.2340
0.2340
0.2340
fixed effects
no
yes
yes
no
yes
yes
specification
OLS
LSDV
OD
OLS
LSDV
OD
sample
Spain
Spain
Spain
EU
EU
EU
period
1955-91
1955-91
1955-91
1980-94
1980-94
1980-94
______________________________________________________________________________
- Notes: Data from Eurostat for 99 regions from the five largest EU countries (Germany, France, UK, Italy and
Spain) and from BBV for the 17 Spanish regions. The Spanish data are available at intervals of generally two (and
sometimes three) years and are not corrected for cross-regional price differences, while the Eurostat data are
annual figures and corrected for differences in purchasing power. In both cases we work with relative income per
capita, that is, income is normalized by its contemporaneous sample average.
Table 3 summarizes the results of the exercise. In the Spanish case, the estimated rate of
unconditional convergence is 2.2% and the standard deviation of the implied asymptotic distribution
of relative income per capita (which reflects only the variance of the shocks εrt ) is 0.10 (column [1] of
Table 3). With the LSDV specification, the (now conditional) convergence rate increases almost four-
26
fold to 8% per year15 and more than half of the regional dummies are highly significant. The implied
steady states look a lot like the end-of-sample incomes and the standard deviation of the implied
_
stationary distribution (taking into account the estimated variance of the shocks) is σy = 0.21, which is
quite close to the observed dispersion in the final year of the sample (σy(1993) = 0.20). Finally, the OD
procedure yields an estimate of the convergence parameter which is only slightly smaller than the
previous one and leaves unaltered the dispersion of the estimated regional steady states (see equation
[3]).
As in previous studies, the conditional and unconditional versions of equation (17) tell very
different stories. In the first case the conclusion is that we have pretty much reached the steady state.
Hence, the substantial degree of inequality we observe today is likely to persist indefinitely in the
absence of "structural change." If we believe the restricted equation, however, we can still hope that
regional inequality in Spain will eventually fall to about one half its current level.
The pattern is similar and even more extreme in the case of the second sample of European regions
(equations [4]-[6] in Table 3). The unconditional specification of equation (17) yields an estimate of the
convergence rate of less than 1%. This figure, however, increases to over 25% when we introduce
fixed effects and, surprisingly, rises even further when we use the OD procedure. As in the case of
Spain, moreover, both fixed effects specifications predict that the long-term dispersion of relative
income per capita will be very close to the observed end-of-sample value.
b.- Full circle back to Solow?
The panel results we have just reviewed are rather problematic if we try to interpret them within
the standard neoclassical framework. The first difficulty has to do with the interpretation of the
convergence rate. Solving for the coefficient of capital, α, in the expression that relates the
convergence rate with the parameters of the production function (equation (15) ), we have
β .
(18) α = 1 g+n+δ
Maintaining our previous assumptions about the rest of the parameters on the right-hand side of
equation (18) (and assuming that the regional dummies adequately capture differences in investment
shares and rates of population growth), the convergence rate we have estimated for the EU regions
(see Table 3) implies a negative value of α, while the estimate for the Spanish sample would leave us,
under the most "favourable" assumption about the value of δ, with a value of α around 0.20. Hence,
our new estimates of the convergence rate take us back, in the best of cases, to the old-fashioned
Solow model with narrowly defined capital -- and often lead to non-sensical results, such as a
negative capital share.
A second problem with similar implications is that panel estimates of the neoclassical model tend
to attribute most of the observed variation in productivity across economies to the country or regional
15 This figure is significantly higher when we work with output per employed worker rather than income per
capita.
27
dummies (i.e. to unknown factors that affect technical efficiency, rather than to differences in factor
stocks) -- a result that says very little in favour of the model's explanatory power. The problem is
illustrated in Figure 6, where I show the fraction of the productivity differential with the sample
average that is explained by cross-country differences in stocks of physical and human capital per
worker for a sample of 21 OECD countries in 1990. When we use the estimates of the production
function parameters obtained by Islam (1995) and Caselli et al (1996), factor stocks account only for
between one tenth and one third of observed productivity differentials.
Figure 6: Fraction of the productivity differential with the sample average explained by
differences in per worker factor endowments in a typical OECD country in 1990
0,6
0,5
0,4
0,3
0,2
0,1
0
Caselli et al
Islam
D&D
- Notes: The weight of factor stocks in the productivity differential is obtained by regressing the contribution of
physical and human capital to log output per worker on log output per worker, with both variables expressed as
differences from sample averages. The assumed coefficients of physical and human capital are 0.373 and 0.271
respectively in de la Fuente and Doménech (D&D) (for a OECD sample), 0.305 and 0.000 for Islam (for an OECD
sample) and 0.107 and 0.00 for Caselli et al (for a sample of 97 countries). The last two authors obtain negative
coefficients for human capital when this variable is included, so we have taken their estimates for the standard
Solow model without human capital. The data on factor stocks are taken from de la Fuente and Doménech (2000).
In a very real sense, these results -- together with the frequent loss of significance of human capital
indicators in panel growth equations-- take us back to 1957, right after the discovery of the Solow
residual, and negate much of what we thought we had learned since then. While it now arises in a
cross-section rather than a time-series setting, the problem is essentially the same one: we cannot
really explain why output varies across time or space in terms of the things we think are important
and know how to measure.
There have been some attempts in the literature to get us out of this corner, but most of them have
not been particularly convincing. Islam (1995) tries to rescue human capital as a determinant of the
level of technological development (which is presumably what is being captured by the country
dummies) by observing that the fixed effects are highly correlated with standard measures of
educational achievement. The argument, however, merely sidesteps the problem: we know that
28
human capital variables work well with cross-section data, but if they really had an effect on the level
of technical efficiency they should be significant when entered into the panel equation. Taking a
different line, Caselli et al (1996) are quite willing to ditch human capital and would settle for the old
fashioned Solow model, but their estimated convergence rate is too high for even that. To rationalize
their results, they turn to some unspecified open-economy version of the standard neoclassical model.
The problem is that, although such a model could indeed generate very fast unconditional
convergence, this should work largely through factor flows. Hence, once we condition on investment
and population growth rates, as Caselli et al do, the estimated convergence rate should reflect only the
characteristics of the technology and would therefore imply an unreasonably low share of capital.
c.- Making sense of fast convergence
Growth economists have spent more than forty years slowing chipping away at the Solow
residual, largely by attributing increasingly larger chunks of it to investment in human capital and
other intangible assets. A few years ago we were reasonably certain that this was the way to go. But
an increasing number of studies seem to be telling us that the effect of these variables on productivity
vanishes when we turn to what seem to be the appropriate econometric techniques for the purpose of
estimating growth equations.
Should we take these results at face value? Before we do so and abandon the only workable
models we have so far, it seems sensible to search for some way to reconcile recent empirical findings
with some kind of plausible theory. In this section, I will argue that this can be done -- at least to some
extent. My argument is essentially that a more reasonable interpretation of the extremely high
convergence rates obtained in recent studies is that, if we have correctly estimated the relevant
parameter (and we may not), then convergence is much too fast to be simply the result of diminishing
returns to scale. This observation points to two complementary lines of research. The first one
proceeds by identifying plausible mechanisms that may help account for rapid convergence and
incorporating them into theoretical and empirical models. The second asks whether panel
specifications of growth equations do in fact yield estimates of the relevant parameter.
Starting with the second line of research, Shioji (1997a and b) and de la Fuente (1998a) provide
some evidence that panel estimates of the convergence rate may tell us very little about the speed at
which economies approach their steady states (and therefore about the degree of returns to scale in
reproducible factors) -- essentially because these estimates are likely to capture short-term
adjustments around trend rather than the long-term growth dynamics we are really interested in.
Both authors show that correcting for the resulting bias in various ways brings us back to convergence
rates that are broadly compatible with sensible theoretical models.
On the first issue, de la Fuente (1995, 1996) and de la Fuente and Doménech (2000) estimate a
further extension of the neoclassical model that allows for cross-country differences in total factor
productivity and for a process of technological catch-up and show that technological diffusion can go
29
a long way towards explaining rapid convergence across countries and regions. 16,17 It follows that
fast convergence does not require us to abandon the broad concept of capital we have so laboriously
developed over the last decades. In fact, the parameters of the aggregate production functions
estimated by these authors at the regional and national level are not far from those obtained by
Mankiw, Romer and Weil (1992) and suggest, in particular that educational investment plays a crucial
role in growth that may not be apparent in previous studies in part because of data deficiencies. On
the other hand, these studies do show that TFP differences across countries and regions are
substantial and account for around half of observed productivity differences in the OECD in recent
years (see Figure 6), a result that is broadly consistent with the findings of Klenow and Rodríguez
(1997) on the explanatory power of the extended neoclassical model. These results highlight the
importance TFP dynamics as a crucial determinant of the evolution of productivity, a subject brought
up recently by Prescott (1998), while retaining a significant role for differences in factor stocks as
sources of income differentials across economies.
7.- Summary and conclusions
In this paper I have reviewed the recent literature on growth and convergence. The first part of the
paper focused on theoretical issues. After discussing the main convergence and divergence
mechanisms identified in growth theory, I have developed a descriptive model that incorporates the
most important such mechanisms and illustrates their implications for the dynamic of the distribution
of income across countries and regions.
In the rest of the paper I have developed a framework for the empirical analysis of growth,
summarized some of the main results of this literature and discussed their theoretical implications. In
the current state of the literature the conclusions we can draw must necessarily remain rather
tentative. Practically all existing studies on the subject find clear evidence of some sort of convergence
both across countries and across regions. These findings allow us to reject with a fair degree of
confidence a series of recent models in which the assumption of increasing returns generates an
explosive behaviour of the distribution of income across economies that cannot be found in the data.
Many of the results we have reviewed are consistent with an extended neoclassical model built
around an aggregate production function that includes human capital as a productive input. Indeed,
such findings seem to have motivated a sort of neoclassical revival that came close to becoming the
conventional wisdom in the literature just a few years ago.
Recently, discussion has livened up again as a result of a series of studies that, using panel data
techniques, turned up rather discouraging results that suggested, in particular, that educational
16 Dowrick and Nguyen (1989) also investigate the quantitative importance of technological catch-up as a
convergence factor, but their empirical specification makes it difficult to disentangle this effect from the
neoclassical convergence mechanism. Helliwell (1992), Coe and Helpman (1995) and Engelbrecht (1997) provide
additional evidence on technological diffusion.
17 There is also some evidence that a significant part of what appears to be TFP convergence at the aggregate
level is in fact due to factor reallocation across sectors. See for instance Paci and Pigliaru (1995), de la Fuente
(1996b), Caselli and Coleman (1999) and de la Fuente and Freire (1999).
30
investment was not productive and that the bulk of productivity differences across countries or
regions has little to do with differences in stocks of productive factors. In my opinion, this has been
largely a false alarm, but it has been useful in shaking up what was probably an exaggerated
confidence in our ability to explain why some countries or regions are richer than others with an
extremely simple model, and in directing researchers' attention to the determinants of technological
progress and to some of the difficult econometric issues involved in the estimation of growth models.
31
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35